mike zaletel station qtensor networks and entanglement spectroscopy mike zaletel station q maglab...
TRANSCRIPT
![Page 1: Mike Zaletel Station QTensor networks and entanglement spectroscopy Mike Zaletel Station Q MagLab Theory Winter School 2015 review of MPS and many of the figures in this talk: Kjäll,](https://reader034.vdocument.in/reader034/viewer/2022050611/5fb22585fb626d58846bd987/html5/thumbnails/1.jpg)
Tensor networks and entanglement spectroscopy
Mike Zaletel Station Q
MagLab Theory Winter School 2015review of MPS and many of the figures in this talk: Kjäll, Zaletel, Mong, Bardarson & Pollmann 2012
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• Day 1: Introduction to tensor network numerics • Entanglement and the Schmidt decomposition • 1D: the matrix product state ansatz
• DMRG • 2D: the ‘tensor network’ ansatz
• Dimer & RVB wavefunctions • Open problems
• Day 2: Entanglement spectroscopy: detecting emergent anyons in numerics
Outline
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Goal:
Find an unbiased method for numerically calculating the low energy properties of any local (perhaps frustrated) quantum Hamiltonian in a time which is polynomial in the system size (or independent of system size with translation invariance).
Some amusing cold water first:[David Pérez-Garcia, Toby Cubitt & Michael Wolf]
[from David Pérez-Garcia]
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The storage problem
""# · · · "
110 · · · 1Classical:
{L
)
S = log2(2L) = L)
Information linear in system size
Quantum:
{ i}(floating points)
) S ⇠ 4 · 8 · 2L
Information exponential in system size (limits exact-diagonalization)
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Quantum compression?
My thesis actually contains surprisingly little information…
Hground states
How big is the important space?
We are interested in states which have low energy for local Hamiltonians
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Hground states
Hokey Estimate I
Parameterize space of ground states via space of local Hamiltonians:
H =LX
i=1
Hi
S / L
HiFinite info for each , so
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Estimate II: the ‘convenient illusion of Hilbert space’
The Setup:
from Poulin, et al., 2011
Start in a product state: |t = 0i = ⌦Ln=1 |"i
Time evolve under an arbitrary k-body Hamiltonian: H(t)
t ⇠ poly(L)After any time we can only access a fraction
of the many-body Hilbert space
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#s
If A & B are uncorrelated (not-entangled), there is a special basis in which
#s
More generally, there is a special basis - the Schmidt basis in which
#s
A B
Schmidt decomposition
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#s
The “entanglement entropy:”
When
we can keep only important contributions and compress the state!
A B
Schmidt compression
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A qubit of entanglement
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Example: 1D transverse field Ising modelL {L
...
A B
...
{Cut length 2L chain in half:
Slightly perturbed from QCP
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The Area Law
AB
[Srednicki]
Proven in 1D for gapped states [Hastings 2005]
Mild violations for certain critical systems (1+1 CFT, Fermi surfaces…)
Ground states:
Volume law expected at finite energy density (eigenstate thermalization)
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The Area Law:
A B
for D = 1 spin chain
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The MPS Ansatz
ij =�X
↵
Ai↵�↵B
j↵
...
A B
...|iiA
Still giant Hilbert space: half the chain
A↵
i
= Ai↵
Penrose graphical notation:Step 1:
cut state in half
A B
i j
↵
i j=
Schmidt coefficients 1 bond to right
...
A B
...
|iiA
’
|j0iB0|piStep 2: Split off 1 site from the right
p j0
B↵
B↵
j
= =
Bj↵ =
�X
�=1
�p↵��
0�B
0j0�
↵B’
�
p j0
�
[Fannes et al. 1992; Östlund & Rommer 1995]
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A↵
i
B’�
p j0i p j0=
The MPS Ansatz
�
Step 3: repeat!
� � � � � � �������� � � � �=
ci1,i2,...,iN =X
↵1,↵2,...,↵N
�[1]i1↵1
�[1]↵1�[2]i2↵1↵2
�[2]↵2
. . .�[N�1]↵N�1
�[N ]iN↵N
Compressed L-site wavefunction into L tensorsd · � · �
�
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Exact diagonalization: ⇠ O(e↵L)
⇠ O(�3)MPS:
MPS: Computing observables
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• Local expectation values
• Correlation functions
• Correlation length: Second largest eigenvalue of the transfer matrix
MPS: Computing observables� �
� ��
�
�
�� �
�
�
� � � �
Simplification Rule: [Vidal 2007]
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DMRG : Density Matrix Renormalization Group
Given , how do we find good a MPS approximations to the g.s.?
MPS:
Minimize
Non-linear minimization problem
1. Hold all tensors fixed but those at site j 2. Solve quadratic problem at site j 3. Move on to site j + 1; repeat
Strategy:
[White 1992; McCullough 2008]
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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Screws up MPS structure!
M =
0
@1 0 0�z
0 0g�
x
�z
1
1
ATransverse Field Ising:
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
For local-ish Hamiltonians, generalize MPS to Matrix Product Operator (MPO)
=
[Verstraete, Porras & Cirac 2004; Murg 2008]Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
![Page 21: Mike Zaletel Station QTensor networks and entanglement spectroscopy Mike Zaletel Station Q MagLab Theory Winter School 2015 review of MPS and many of the figures in this talk: Kjäll,](https://reader034.vdocument.in/reader034/viewer/2022050611/5fb22585fb626d58846bd987/html5/thumbnails/21.jpg)
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
R
ΓB ΛB
(Γ )*Λ
ΓB ΛB
(Γ )B * B BΛ
ΓA ΛA
(Γ )A *ΛA B
���������
L
ΓBΛA
(Γ )*Λ
ΓAΛA
(Γ )A * BAΛ
ΓAΛB
(Γ )A *ΛB A
���������
MR0
L0
M M
M M M
ΓA ΛA ΓB ΛBΛB(i)
(ii)
Θ
ΛA(iii) X Y~
SVD
Θ Θ~
U
Θ~
(iv) ΓA ΛA ΓB ΛBΛB ~ ~ ~
( )-1ΛAX Y~ΛB ΛB
ΛB -1
ΛB ( )
Variational Wavefunction:
2 sites + L / R Schmidt statesOrthonormal basis for
Focus on two sites
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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ΓA ΛA ΓB ΛBΛB(i)
(ii)
ΛA(iii)
X Y
(iv)ΓA ΛA ΓB ΛBΛB
~
~ ~ ~
( )-1
SVD
Θ~
ΛAX Y~ΛB ΛB
ΛB -1
ΛB ( )
=Θ~
E0L R
L
R
~
~
ΓAΛB ~
(Γ )A *ΛB ~
ΓBΛB~
(Γ )B *ΛB~
Θ~
M M
Θ
(v)
Lower the energy by finding the ground state of effective Hamiltonian (Lanczos, etc.):
This is where you burn CPU hours:
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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ΓA ΛA ΓB ΛBΛB(i)
(ii)
ΛA(iii)
X Y
(iv)ΓA ΛA ΓB ΛBΛB
~
~ ~ ~
( )-1
SVD
Θ~
ΛAX Y~ΛB ΛB
ΛB -1
ΛB ( )
=Θ~
E0L R
L
R
~
~
ΓAΛB ~
(Γ )A *ΛB ~
ΓBΛB~
(Γ )B *ΛB~
Θ~
M M
Θ
(v)
Bring ansatz back to MPS form
A B
�
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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ΓA ΛA ΓB ΛBΛB(i)
(ii)
ΛA(iii)
X Y
(iv)ΓA ΛA ΓB ΛBΛB
~
~ ~ ~
( )-1
SVD
Θ~
ΛAX Y~ΛB ΛB
ΛB -1
ΛB ( )
=Θ~
E0L R
L
R
~
~
ΓAΛB ~
(Γ )A *ΛB ~
ΓBΛB~
(Γ )B *ΛB~
Θ~
M M
Θ
(v)
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
Update L / R environments
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
�������� M M M M M
�������� M M M M M(b) � � � � � � ����
(a)
���� � � � �
�������� M M M M M(c) � � � � � � �������� � � � �
� � � � � ���������
� � � �* * * * *
“Sweep” until convergence
Review of algorithm: Kjäll, Zaletel, Bardarson, Mong & Pollmann 2012, 1212.6255
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Algorithm works unchanged on an infinitely long system with periodic unit cell: “iDMRG”
Complexity: length / unit cell = L
Comments
CPU:
RAM:
(holding Hamiltonian fixed)
2D:
1D CFT:
1D gapped:
[McCullough 2008]
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Time evolution....
U U UU Ut
ΓA ΛA ΓB ΛB ΓAΛA ΓBΛB ΓAΛA....
Trotter-decompose U(dt) into 2-site gates:
Dynamical structure factor:
TEBD [Vidal 03]
�
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Time evolution
Experiments of Coldea, et al.: 1D TFI perturbed by order parameter
Near QCP: masses of emergent excitations root lattice of E_8
[from Kjäll 2011]
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2D DMRG: The Kludge (alias - snake)
Order the 2D lattice into 1D chain with longer-range interactions
Entanglement scales with circumference:
DMRG Exact DiagonalizationComplexity:
[from Stoudenmire 2011]
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Fractional quasiparticles in the fractional quantum hall effect
[Zaletel, Mong, Pollmann 2012]
[Yan, Huse, White 2010]
Frustrated magnetism & Spin-liquids on cylinders
2D DMRG
Works if you are lucky (i.e., near thermodynamic limit on small cylinders)
More on measuring topological order in these studies tomorrow
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2D Tensor network: The Hope
=
[Verstraete & Cirac, 2004]
[from Roman Orus, 1306.2164 Great review!]
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Example: dimer model
Kagome NN dimer covering:
[from Yejin Huh, 2011]
[RVB PEPs examples: Schuch, et al. 2012]
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Using different topology than square TN: but you can always regroup things
to turn it into “standard” form
p = 0: no dimer p = 1: dimer
The constraint (no physical index)Glue presence of dimer to “virtual” index
TN for dimers
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[from Stoudenmire 2011]
Square-lattice J1
Why is Kagome still being studied with snakes?
Finding the 2D TN is hard!
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Why is Kagome still being studied with snakes?
Calculating expectation value in MPS exactly: linear complexity in size Calculating expectation value in 2d TN exactly: exponential complexity in size
*** Unsolved problem 1: what is the right way to approximately calculate observables in a 2D TN? ***
DMRG: it works. complexity 2D TN: algorithms proposed, but not fully understood what the nature of the approximations is. complexity
*** Unsolved problem 2: what is the right way to find a 2D TN given H? ***
1D MPS: represents gapped states of local H 2D TN: not known (certain things can’t be: fermi surface)
*** Unsolved problem 0: which phases of matter can be represented by finite dimensional 2D TN? ***
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Thanks!
Mike Zaletel Station Q
MagLab Theory Winter School 2015